Combining Texts

All the ideas for 'Demonstratives', 'Is Justified True Belief Knowledge?' and 'Mathematics without Numbers'

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6 ideas

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
8698 Modal structuralism says mathematics studies possible structures, which may or may not be actualised [Hellman, by Friend]
     Full Idea: The modal structuralist thinks of mathematical structures as possibilities. The application of mathematics is just the realisation that a possible structure is actualised. As structures are possibilities, realist ontological problems are avoided.
     From: report of Geoffrey Hellman (Mathematics without Numbers [1989]) by Michčle Friend - Introducing the Philosophy of Mathematics 4.3
     A reaction: Friend criticises this and rejects it, but it is appealing. Mathematics should aim to be applicable to any possible world, and not just the actual one. However, does the actual world 'actualise a mathematical structure'?
9557 Statements of pure mathematics are elliptical for a sort of modal conditional [Hellman, by Chihara]
     Full Idea: Hellman represents statements of pure mathematics as elliptical for modal conditionals of a certain sort.
     From: report of Geoffrey Hellman (Mathematics without Numbers [1989]) by Charles Chihara - A Structural Account of Mathematics 5.3
     A reaction: It's a pity there is such difficulty in understanding conditionals (see Graham Priest on the subject). I intuit a grain of truth in this, though I take maths to reflect the structure of the actual world (with possibilities being part of that world).
10263 Modal structuralism can only judge possibility by 'possible' models [Shapiro on Hellman]
     Full Idea: The usual way to show that a sentence is possible is to show that it has a model, but for Hellman presumably a sentence is possible if it might have a model (or if, possibly, it has a model). It is not clear what this move brings us.
     From: comment on Geoffrey Hellman (Mathematics without Numbers [1989]) by Stewart Shapiro - Philosophy of Mathematics 7.3
     A reaction: I can't assess this, but presumably the possibility of the model must be demonstrated in some way. Aren't all models merely possible, because they are based on axioms, which seem to be no more than possibilities?
13. Knowledge Criteria / A. Justification Problems / 2. Justification Challenges / b. Gettier problem
8886 Being a true justified belief is not a sufficient condition for knowledge [Gettier]
     Full Idea: The claim that someone knows a proposition if it is true, it is believed, and the person is justified in their belief is false, in that the conditions do not state a sufficient condition for the claim.
     From: Edmund L. Gettier (Is Justified True Belief Knowledge? [1963], p.145)
     A reaction: This is the beginning of the famous Gettier Problem, which has motivated most epistemology for the last forty years. Gettier implies that justification is necessary, even if it is not sufficient. He gives two counterexamples.
19. Language / C. Assigning Meanings / 10. Two-Dimensional Semantics
14894 Indexicals have a 'character' (the standing meaning), and a 'content' (truth-conditions for one context) [Kaplan, by Macią/Garcia-Carpentiro]
     Full Idea: Kaplan distinguished two different semantic features of indexical expressions: a 'character' that captures the standing meaning of the expression, and a 'content' that consists of their truth-conditional contribution in particular contexts.
     From: report of David Kaplan (Demonstratives [1989]) by Macią/Garcia-Carpentiro - Introduction to 'Two-Dimensional Semantics' 1
     A reaction: This seems so clearly right that there isn't much to dispute. You can't understand the word 'I' or 'now' if you don't understand both its general purpose, and what it is doing in a particular utterance. But will this generalise to other semantics?
14700 'Content' gives the standard modal profile, and 'character' gives rules for a context [Kaplan, by Schroeter]
     Full Idea: Kaplan sees two aspects of meaning, the 'content', reflecting a thing's modal profile, which is modelled by standard possible worlds semantics, and 'character', giving rules for different contexts. Proper names have constant character; indexicals vary.
     From: report of David Kaplan (Demonstratives [1989]) by Laura Schroeter - Two-Dimensional Semantics 1.1.1
     A reaction: This gives rise to 2-D matrices for representing meaning, and the possible worlds are used twice, for evaluating meaning and then for evaluating context of use. I've always been struck by the two-dimensional semantics of passwords.